Sequences and Series Review

A sequence is an ordered list of numbers that follow a specific pattern or rule.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

The nth term of an arithmetic sequence can be found using the formula: $$a_n = a_1 + (n-1)d$$ , where $$a_1$$ is the first term and $$d$$ is the common difference.

The nth term of a geometric sequence can be found using the formula: $$a_n = a_1 \times r^{(n-1)}$$ , where $$a_1$$ is the first term and $$r$$ is the common ratio.

The sum of the first n terms of an arithmetic series can be calculated using the formula: $$S_n = \frac{n}{2} (a_1 + a_n)$$ .

The sum of the first n terms of a geometric series can be calculated using the formula: $$S_n = a_1 \frac{1 - r^n}{1 - r}$$ , where @@r eq 1@@.

A convergent geometric series is a geometric series where the absolute value of the common ratio is less than 1, leading to a finite sum.

The sum of an infinite convergent geometric series can be calculated using the formula: $$S = \frac{a_1}{1 - r}$$ , where $$|r| < 1$$ .

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. To determine if it is geometric, check if the ratio of consecutive terms is constant.